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>] EANDOM FLIGHTS IN ONE, TWO, OR THREE DIMENSIONS 623 these examples the procedure will be understood. When n is even, Differentiate (59) (n 2) times, thus obtaining dP \ n 2 r°° d(K 1M = (1)2- sin rao sin91 lx. .........(63) dr J Trln Jo a? hich smnlx is replaced by the series containing cos nix, cos (n 2) lx, ... ending with a constant term. When this is multiplied by sin rx, we sines of (r ± nl)x, {r + (n 2) 1} cc, ... sin roo, and the integration can be ted. Over the various ranges of 2Z the values are constant, but they .ge discontinuously when r is an even multiple of I. The actual forms r.Pn/dr can then be found, as already exemplified, by working backwards r>nl, where all derivatives vanish, and so determining the constants of ^ration as to maintain continuity throughout. These forms are in all 3 algebraic. tVhen n is odd, we differentiate (??. 3) times, thus obtaining a form lar to (60) where n = 3. A similar procedure then shows that the result rn.es constant values over finite ranges with discontinuities* when r is an multiple of I. On integration the forms for dPn/dr are again algebraic. '. have carried out the detailed calculation for n = 6. It will suffice to :d the principal results. For the values of fa. J :nd for the various ranges : (r<2Z), -20; (2Z<r<4Z), +10; (4Z<r<6Z), -2; (6Z < r), 0. on integration for r dr ) ' .(64) 5r4 ~ (2Z - 6Zr» + 30/2r2 - 56ZV + (4Z - 6Z)f - + 2Zr» - 18ZV8 + 72Z'r - 108Z4, 1^1 (r > 6Z) 0. ), since the integrals There are, however, no discontinuities in the value of u . 9(r-nl)a f00 dx . 9{r-(n-2)l}x ,. , . ., .. , . ,. sm2 -, I 5 sin2-----i-, etc., which appear m the result when n is odd, i £ j o x & intintious for all values of r (cf. the solution for - on p. 621). The result for (4Z-61) maybe written --^ (6Z-?)*. And in general, when (?z- 2 ,-----y (nl-r)*-2, whether n be even or odd. W. F. S.] II f(:4 appears to have been devised to meet the case when s = 0 and the integral,, though finite, does not converge to a definite value when x = <x>. If, however, ,s < -1, or <0, respectively, according as the cosine factors in (38) do or do not produce a constant term, the integral (38) has been shown to be finite; it is -also convergent; and the integrals obtained by omitting before each successive differentiation the factor to be differentiated, viz.': cos (%ir-rx-lqir) where g<p, are also finite (cf. Todhifnter's Integral Calculus, 1889, Arts. 214, 284). In these circumstances it would appear that-_(38) is itself valid, and that it is unnecessary to consider the integral obtained by "the next following operation" (s =p + f - %ri). It would seem then that the above considerations are sufficient to justify the differentiation by which 0,t is obtained (jp = 0, s- -f), and a fortiori that for <fe (# = 0, s=-2), etc. W..IY.S.]elative index is then much restricted, the